Integrability of Kepler Billiards at Zero-Energy

TL;DR

This paper proves that a zero-energy Kepler billiard with certain invariant structures must be inside an elliptical boundary, extending a known theorem via Levi-Civita transformation.

Contribution

It demonstrates that under specific conditions, the billiard system is necessarily confined within an ellipse, generalizing the Bialy-Mironov theorem through Levi-Civita transformation.

Findings

System is confined inside an ellipse under given conditions.

Invariant curve of 2-periodic orbits implies elliptical boundary.

Theorem extends Bialy-Mironov result via Levi-Civita transformation.

Abstract

We consider a Kepler billiard with zero-energy in the plane defined inside a smooth closed connected simple curve which intersects all focused parabola at at most two points. {We show that} if has an invariant curve consisting of $2$-periodic orbits and there exists a $C^{1}$-first integral with non-vanishing gradient in the region between the invariant curve and the boundary curve, then the system is defined actually inside an ellipse with the Kepler center occupying one of the foci. This statement is obtained as a simple ``translation'' of the theorem of Bialy-Mironov with Levi-Civita transformation.